Quadratic expression of Kuznets curve

[Murielle Fuchs]

Good morning,

I am trying to run a regression for a waste Kuznets curve (panel data). I used the quadratic term of GDP as suggested by most authors but this variable is then omitted because of collinearity. I understand that the multicollinearity comes from its relationship with the GDP variable, but do not understand how other authors fixed this problem.

Would anyone be able to find a solution?

gen GDP2t = GDP * GDP
gen lnGDP2t = ln(GDP2t)
xtreg lnWaste lnGDP lnGDP2t lnUrban lnHhsize lnEducated

[Nick Cox]

In general, the logarithm of a square is twice the logarithm of the original, just as the logarithm of 10^6 is twice the logarithm of its root 10^3. (If the base is 10, then 6 = 2 x 3, but using any base you could get a similar result.)

So, collinearity within your variables is perfect.

I imagine that what you seek is more likely to be

gen lnGDP2 = lnGDP^2  
xtreg lnWaste lnGDP lnGDP2 lnUrban lnHhsize lnEducated

[Murielle Fuchs]

Thank you very much Nick, I indeed had a mistake in my maths.